Almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice versa. Almost complex structures have important applications in symplectic geometry.
The concept is due to Ehresmann and Hopf in the 1940s.
Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of degree (1, 1) such that J 2 = −1 when regarded as a vector bundle isomorphism J : TM → TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold.
If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. If J 2 = −1 then det(J)2 = (−1)n. But if M is a real manifold, then det(J) is a real number- thus n must be even if M has an almost complex structure. One can show that it must be orientable as well.
An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a (1, 1) rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold M is equivalent to a reduction of the structure group of the tangent bundle from GL(2n, R) to GL(n, C). The existence question is then a purely algebraic topological one and is fairly well understood.
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